Abstract: We prove that if is a flat local homomorphism, is Cohen-Macaulay and -injective, and and share a weak test element, then a tight closure analogue of the (standard) formula for depth and regular sequences across flat base change holds. As a corollary, it follows that phantom depth commutes with completion for excellent local rings. We give examples to show that the analogue does not hold for surjective base change.

[Hun96]Craig
Huneke, Tight closure and its applications, CBMS Regional
Conference Series in Mathematics, vol. 88, Published for the
Conference Board of the Mathematical Sciences, Washington, DC; by the
American Mathematical Society, Providence, RI, 1996. With an appendix by
Melvin Hochster. MR 1377268
(96m:13001)